Module 3 - Equities & Currencies

In module three, we will explore the importance of the Black- Scholes theory as a theoretical and practical pricing model which is built on the principles of delta heading and no arbitrage. You will learn about the theory and results in the context of equities and currencies using different kinds of mathematics to make you familiar with techniques in current use.

Black-Scholes Model
  • The assumptions that go into the Black-Scholes equation
  • Foundations of options theory: delta hedging and no arbitrage
  • The Black-Scholes partial differential equation
  • Modifying the equation for commodity and currency options
  • The Black-Scholes formulae for calls, puts and simple digitals
  • The meaning and importance of the Greeks, delta, gamma, theta, vega and rho
  • American options and early exercise
  • Relationship between option values and expectations
Martingale Theory - Applications to Option Pricing
  • Computing the price of a derivative as an expectation
  • Introducing the change of measures
  • The role of self-financing trading strategy
  • The Fundamental Asset Pricing Formula
  • Quick PDE via the Feynman-Kac formula
  • The Black-Scholes call option problem
  • Black's formula for options on futures
Option Pricing Models: Connecting the Dots
  • One-period binomial model: the risk-neutral measure and equivalent martingale measure are the
    same
  • The Fundamental Asset Pricing Formula in one and multi-period binomial model
  • Derive the Black-Scholes PDE from one-period binomial model
  • The multi-period binomial model yields the Black-Scholes formulae
  • Complete markets and incomplete markets defined
  • How the Black-Scholes is a model of the complete market
  • Finding a martingale measure implies: there is no arbitrage
Intro to Numerical Methods
  • Simulating discrete paths of the stochastic asset price
  • The justification for pricing by Monte Carlo simulation
  • Grids and discretization of necessary partial derivatives of an option price
  • Backward, forward and central differences
  • The explicit finite-difference method
  • The explicit finite-difference: general conditions and Von Neumann’s stability
Exotic Options
  • How to classify exotic options according to important payoff features
  • Time dependence (Bermudian options)
  • Path dependence and embedded decisions
  • Dimensionality and pricing methods
  • Asian option with continuous and discrete sampling
Understanding Volatility
  • The many types of volatility
  • What the market prices of options tell us about volatility
  • The term structure
  • Intro to volatility skews and smiles
  • Volatility arbitrage: should you hedge using implied or actual volatility?
Further Numerical Methods
  • Implicit finite-difference
  • Crank-Nicolson scheme and its matrix form
  • Douglas schemes
  • American-style exercise
  • Solving two-factor models with the explicit difference scheme
Things They Don't Teach You In School

Simple ideas that can have profound importance
 

  • Lessons from the mathematics of gambling. "Happiness"
  • Jensen's Inequality
  • Discrete hedging and transaction costs
  • Extreme markets
  • Is volatility important?
     
Common Quant Mistakes

A few of the things that quants get wrong:
 

  • Sensitivity to parameters
  • Correlation
  • Lack of diversification/size of trades
  • Reliance on dynamic hedging (arguments), risk-neutral vs real
  • Feedback
  • Supply and demand
Advanced Volatility Modeling in Complete Markets
  • The relationship between implied volatility and actual volatility in a deterministic world
  • The difference between 'random' and 'uncertain'
  • How to price contracts when volatility, interest rate and dividend are uncertain
  • Non-linear pricing equations
  • Optimal static hedging with traded options
  • How non-linear equations make a mockery of calibration
FX Options
  • Size and importance of the FX and FX options market
  • How FX has developed into the largest global market
  • Current uses of FX options
  • Volatility surface and out of money options
  • Pricing of simple FX options and those replicated from vanilla calls and puts
  • Path-dependent FX options American and Bermudan
  • Risk management and basic delta hedging

Lecture order and content may occasionally change due to circumstances beyond our control. However, this will never affect the quality of the program.

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